Physics-integrated neural differentiable modeling for immersed boundary systems

Chenglin Li; Hang Xu; Jianting Chen; Yanfei Zhang

cs.LGMar 17, 2026

Physics-integrated neural differentiable modeling for immersed boundary systems

Authors: Chenglin Li, Hang Xu, Jianting Chen, Yanfei Zhang

Organizations: State Key Lab of Ocean Engineering, School of Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China · State Key Laboratory of Maritime Technology and Safety, Shanghai Ship and Shipping Research Institute Co., Ltd, Shanghai 200135, China.

Abstract

Accurately, efficiently, and stably computing complex fluid flows and their evolution near solid boundaries over long horizons remains challenging. Conventional numerical solvers require fine grids and small time steps to resolve near-wall dynamics, resulting in high computational costs, while purely data-driven surrogate models accumulate rollout errors and lack robustness under extrapolative conditions. To address these issues, this study extends existing neural PDE solvers by developing a physics-integrated differentiable framework for long-horizon prediction of immersed-boundary flows. A key design aspect of the framework includes an important improvement, namely the structural integration of physical principles into an end-to-end differentiable architecture incorporating a PDE-based intermediate velocity module and a multi-direct forcing immersed boundary module, both adhering to the pressure-projection procedure for incompressible flow computation. The computationally expensive pressure projection step is substituted with a learned implicit correction using ConvResNet blocks to reduce cost, and a sub-iteration strategy is introduced to separate the embedded physics module's stability requirement from the surrogate model's time step, enabling stable coarse-grid autoregressive rollouts with large effective time increments. The framework uses only single-step supervision for training, eliminating long-horizon backpropagation and reducing training time to under one hour on a single GPU. Evaluations on benchmark cases of flow past a stationary cylinder and a rotationally oscillating cylinder at Re=100 show the proposed model consistently outperforms purely data-driven, physics-loss-constrained, and coarse-grid numerical baselines in flow-field fidelity and long-horizon stability, while achieving an approximately 200-fold inference speedup over the high-resolution solver.

Figures & tables

Figure 1: Schematic of the computational setup and domains. The immersed-boundary (IB) solver is defined on the global fluid domain $\Omega_{f}$ (gray box) with boundaries $\Gamma_{\mathrm{inlet}}$ , $\Gamma_{\mathrm{outlet}}$ , $\Gamma_{\mathrm{upper}}$ , and $\Gamma_{\mathrm{lower}}$ . The surrogate model is evaluated on a truncated subdomain $\Omega^{r}_{f}$ (blue box) bounded by $\Gamma^{r}_{\mathrm{inlet}}$ , $\Gamma^{r}_{\mathrm{outlet}}$ , $\Gamma^{r}_{\mathrm{upper}}$ , and $\Gamma^{r}_{\mathrm{lower}}$ . A prescribed inflow $u=u_{\mathrm{in}}$ is imposed at $\Gamma_{\mathrm{inlet}}$ , and free-slip conditions are enforced on $\Gamma_{\mathrm{upper}}$ and $\Gamma_{\mathrm{lower}}$ .

Figure 2: Schematic of the proposed physics-integrated differentiable neural network. Trainable neural modules (ConvResNet blocks) are shown in green, while non-trainable physics operators (the PDE operator and the multi-direct-forcing IBM module) are highlighted in yellow. Grey dashed boxes enclose the inputs, outputs, and intermediate physical variables. Data propagate along the black arrows. The PDE-operator substep is iterated $N$ times, and up-/down-sampling bridges the resolutions between the physics and neural components.

Figure 3: Comparison of spatio-temporal vorticity predictions by different neural models for flow past a stationary circular cylinder at $(Re=100,\omega=0)$ .

Figure 4: Comparison of structural force predictions by different neural models for flow past a stationary circular cylinder at $(Re=100,\omega=0)$ . The left column shows the time histories of the drag coefficient $C_{D}$ , and the right column shows the lift coefficient $C_{L}$ . The gray-shaded interval denotes the training window, whereas the white interval corresponds to out-of-distribution time extrapolation beyond the training horizon. The black solid line represents the ground truth, and the blue curve corresponds to the prediction of the proposed model.

Figure 5: Stepwise rollout relative error $\epsilon^{\,t}$ for flow past a stationary circular cylinder at $Re=100$ and $\omega=0$ , defined in Eq. ( 25 ). The metric is computed as the average of the relative $L_{2}$ errors of $u$ , $v$ , and the structural response vector $\mathbf{F}$ at each rollout step. The gray-shaded region indicates the training window, and the white region indicates temporal extrapolation beyond the training horizon.

Figure 6: Comparison of spatio-temporal vorticity predictions by different neural models for flow past a rotationally oscillating circular cylinder at $(Re=100,\omega=4)$ .

Figure 7: Comparison of structural force predictions by different neural models for flow past a rotationally oscillating circular cylinder at $(Re=100,\omega=4)$ . The left column shows the time histories of the drag coefficient $C_{D}$ , and the right column shows the lift coefficient $C_{L}$ . The gray-shaded interval denotes the training window, whereas the white interval corresponds to out-of-distribution time extrapolation beyond the training horizon. The black solid line represents the ground truth, and the blue curve corresponds to the prediction of the proposed model.

Figure 8: Stepwise rollout relative error $\epsilon^{\,t}$ for flow past a rotationally oscillating circular cylinder at $Re=100$ and $\omega=4$ , defined in Eq. ( 25 ). The metric is computed as the average of the relative $L_{2}$ errors of $u$ , $v$ , and the structural response vector $\mathbf{F}$ at each rollout step. The gray-shaded region indicates the training window, and the white region indicates temporal extrapolation beyond the training horizon.

Figure 9: Average relative error $\bar{\epsilon}$ (averaged over the entire inference horizon) under unseen operating conditions for the rotationally oscillating cylinder. Models are trained on $\omega=\{2,4\}$ and evaluated at $\omega=3$ (interpolation) and $\omega=5$ (extrapolation). Lower values indicate better long-horizon generalization.

Figure 10: Comparison of spatio-temporal vorticity predictions by different neural models for flow past a rotationally oscillating circular cylinder at $(Re=100,\omega=5)$ .

Figure 11: Comparison of structural force predictions by different neural models for flow past a rotationally oscillating circular cylinder at $(Re=100,\omega=5)$ . The left column shows the time histories of the drag coefficient $C_{D}$ , and the right column shows the lift coefficient $C_{L}$ . The gray-shaded interval denotes the training window, whereas the white interval corresponds to out-of-distribution time extrapolation beyond the training horizon. The black solid line represents the ground truth, and the blue curve corresponds to the prediction of the proposed model.

Figure 12: Wall-clock time per learning step for different methods. The neural models are evaluated on the truncated subdomain $\Omega^{r}_{f}$ using an $80\times 40$ grid, whereas the ground-truth CFD solution is computed on the full domain $\Omega_{f}$ using a $717\times 382$ grid.

Figure 13: Training efficiency comparison between the single-step and BPTT-based multi-step training for flow past a stationary circular cylinder at Reynolds number $Re=100$ . The shaded bands represent $\pm\sigma$ .

Training strategy	GPU memory usage (GB)
Single-step training	2.7
BPTT ( $T=3$ )	5.7
BPTT ( $T=5$ )	8.8

Table 1: GPU memory consumption under different training strategies.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

	$\bar{C}_{D}$	$C_{L}^{\prime}$	$St$
Our	1.383	$\pm 0.345$	0.167
Cui et al. [ 47 ]	1.360	$\pm 0.340$	0.167
Uhlmann [ 48 ]	1.453	$\pm 0.339$	0.169
Wang et al. [ 49 ]	1.334	$\pm 0.370$	0.163

Appendix

Table A1: Dimensionless coefficients obtained from the simulation of the flow around a stationary cylinder at

Re=100

, using

D=1

and

dt=0.005

Figure B1: Illustration of transferring staggered-grid variables from a fine mesh to an $n\times n$ coarser mesh via downsampling (an $8\times$ reduction in resolution is used in this study). For each coarse control volume, the face-centered velocity components on the fine grid are area-averaged to obtain the corresponding coarse-grid velocities, while the cell-centered pressures within the fine-grid subcells are averaged to yield the coarse-grid pressure. Fine-grid velocity samples located in the interior of the coarse cell are not retained.

Figure C2: Architectures of the baseline models used for comparison. (a) Purely data-driven model trained with the data loss $\mathcal{L}_{r}$ . (b) Physics loss constrained model, which augments (a) with the physics-consistency term $\mathcal{L}_{\mathrm{phy}}=\mathcal{L}_{\mathrm{IB}}+\mathcal{L}_{m}+\mathcal{L}_{\mathrm{div}}$ and optimizes the total loss $\mathcal{L}_{\mathrm{total}}=\alpha\mathcal{L}_{\mathrm{phy}}+\beta\mathcal{L}_{r}$ .

Explore similar work

Multiscale Physics-Informed Neural Network for Complex Fluid Flows with Long-Range Dependencies
Apr 7, 2026Prashant Kumar, Rajesh RanjanNeural PDE Solvers Physics-Informed ML
Department of Aerospace Engineering, Indian Institute of Technology Kanpur, Kanpur, India 208016
An End-to-End PyTorch Interface for Differentiable PDE Solvers: A RANS Model-Correction Study
May 19, 2026Luca Saverio, Michele Alessandro Bucci, Gianmarco Farro +2PDE Solving Differentiable Optimization
MONHADE · Digital Sciences & Technologies Department, Safran Tech, Magny-Les-Hameaux, 78114, France. · DAAA, ONERA, Institut Polytechnique de Paris, Meudon, 92190, France. +1
A Multi-Resolution Finite-Volume Inspired Deep Learning Framework for Spatiotemporal Dynamics Prediction
Jul 1, 2026Xin-Yang Liu, Xiantao Fan, Jian-Xun WangNeural PDE Solvers PDE Surrogate Modeling
Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN, USA · Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY, USA

Apr 7, 2026J/K move · Enter open · S save

Multiscale Physics-Informed Neural Network for Complex Fluid Flows with Long-Range Dependencies

Prashant Kumar, Rajesh Ranjan

Department of Aerospace Engineering, Indian Institute of Technology Kanpur, Kanpur, India 208016

Neural PDE Solvers Physics-Informed ML PDE Solving Scientific ML Computational Fluid Dynamics

Open paper AskPDF arXiv

Fluid flows are governed by the nonlinear Navier-Stokes equations, which can manifest multiscale dynamics even from predictable initial conditions. Predicting such phenomena remains a formidable challenge in scientific machine learning, particularly regarding convergence speed, data requirements, and solution accuracy. In complex fluid flows, these challenges are exacerbated by long-range spatial dependencies arising from distant boundary conditions, which typically necessitate extensive supervision data to achieve acceptable results. We propose the Domain-Decomposed and Shifted Physics-Informed Neural Network (DDS-PINN), a framework designed to resolve such multiscale interactions with minimal supervision. By utilizing localized networks with a unified global loss, DDS-PINN captures global dependencies while maintaining local precision. The robustness of the approach is demonstrated across a suite of benchmarks, including a multiscale linear differential equation, the nonlinear Burgers' equation, and data-free Navier-Stokes simulations of flat-plate boundary layers. Finally, DDS-PINN is applied to the computationally challenging backward-facing step (BFS) problem; for laminar regimes (Re = 100), the model yields results comparable to computational fluid dynamics (CFD) without the need for any data, accurately predicting boundary layer thickness, separation, and reattachment lengths. For turbulent BFS flow at Re = 10,000, the framework achieves convergence to O(10^-4) using only 500 random supervision points (< 0.3 % of the total domain), outperforming established methods like Residual-based Attention-PINN in accuracy. This approach demonstrates strong potential for the super-resolution of complex turbulent flows from sparse experimental measurements.

Abstract

Explore similar work

Multiscale Physics-Informed Neural Network for Complex Fluid Flows with Long-Range Dependencies

An End-to-End PyTorch Interface for Differentiable PDE Solvers: A RANS Model-Correction Study

A Multi-Resolution Finite-Volume Inspired Deep Learning Framework for Spatiotemporal Dynamics Prediction